Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pellfundgt1 | Structured version Visualization version GIF version |
Description: Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
Ref | Expression |
---|---|
pellfundgt1 | ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 9934 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 ∈ ℝ) | |
2 | eldifi 3694 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℕ) | |
3 | 2 | peano2nnd 10914 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℕ) |
4 | 3 | nnrpd 11746 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℝ+) |
5 | 4 | rpsqrtcld 13998 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘(𝐷 + 1)) ∈ ℝ+) |
6 | 5 | rpred 11748 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘(𝐷 + 1)) ∈ ℝ) |
7 | 2 | nnrpd 11746 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℝ+) |
8 | 7 | rpsqrtcld 13998 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘𝐷) ∈ ℝ+) |
9 | 8 | rpred 11748 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘𝐷) ∈ ℝ) |
10 | 6, 9 | readdcld 9948 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ∈ ℝ) |
11 | pellfundre 36463 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ) | |
12 | sqrt1 13860 | . . . . 5 ⊢ (√‘1) = 1 | |
13 | 12, 1 | syl5eqel 2692 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘1) ∈ ℝ) |
14 | 13, 13 | readdcld 9948 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘1) + (√‘1)) ∈ ℝ) |
15 | 1lt2 11071 | . . . . 5 ⊢ 1 < 2 | |
16 | 12, 12 | oveq12i 6561 | . . . . . 6 ⊢ ((√‘1) + (√‘1)) = (1 + 1) |
17 | 1p1e2 11011 | . . . . . 6 ⊢ (1 + 1) = 2 | |
18 | 16, 17 | eqtri 2632 | . . . . 5 ⊢ ((√‘1) + (√‘1)) = 2 |
19 | 15, 18 | breqtrri 4610 | . . . 4 ⊢ 1 < ((√‘1) + (√‘1)) |
20 | 19 | a1i 11 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < ((√‘1) + (√‘1))) |
21 | 3 | nnge1d 10940 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 ≤ (𝐷 + 1)) |
22 | 0le1 10430 | . . . . . . 7 ⊢ 0 ≤ 1 | |
23 | 22 | a1i 11 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 0 ≤ 1) |
24 | 2 | nnred 10912 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℝ) |
25 | peano2re 10088 | . . . . . . 7 ⊢ (𝐷 ∈ ℝ → (𝐷 + 1) ∈ ℝ) | |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℝ) |
27 | 3 | nnnn0d 11228 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℕ0) |
28 | 27 | nn0ge0d 11231 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 0 ≤ (𝐷 + 1)) |
29 | 1, 23, 26, 28 | sqrtled 14013 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (1 ≤ (𝐷 + 1) ↔ (√‘1) ≤ (√‘(𝐷 + 1)))) |
30 | 21, 29 | mpbid 221 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘1) ≤ (√‘(𝐷 + 1))) |
31 | 2 | nnge1d 10940 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 ≤ 𝐷) |
32 | 2 | nnnn0d 11228 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℕ0) |
33 | 32 | nn0ge0d 11231 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 0 ≤ 𝐷) |
34 | 1, 23, 24, 33 | sqrtled 14013 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (1 ≤ 𝐷 ↔ (√‘1) ≤ (√‘𝐷))) |
35 | 31, 34 | mpbid 221 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘1) ≤ (√‘𝐷)) |
36 | 13, 13, 6, 9, 30, 35 | le2addd 10525 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘1) + (√‘1)) ≤ ((√‘(𝐷 + 1)) + (√‘𝐷))) |
37 | 1, 14, 10, 20, 36 | ltletrd 10076 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < ((√‘(𝐷 + 1)) + (√‘𝐷))) |
38 | pellfundge 36464 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (PellFund‘𝐷)) | |
39 | 1, 10, 11, 37, 38 | ltletrd 10076 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∖ cdif 3537 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 ℕcn 10897 2c2 10947 √csqrt 13821 ◻NNcsquarenn 36418 PellFundcpellfund 36422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-ico 12052 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-numer 15281 df-denom 15282 df-squarenn 36423 df-pell1qr 36424 df-pell14qr 36425 df-pell1234qr 36426 df-pellfund 36427 |
This theorem is referenced by: pellfundex 36468 pellfundrp 36470 pellfundne1 36471 pellfund14 36480 |
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